19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER Methodology for electro mechanical simulation of piezoelectric ceramics

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1 19 th INTRNATIONAL CONR ON ACOUTIC MADRID, 2-7 PTMBR 2007 Methodology for electro mechanical simulation of piezoelectric ceramics PAC: Fx Morcillo López, Miguel Ángel 1 ; Cesteros Morante, Beatriz 1 ; Cordero Izquierdo, Roberto 1 ; Hidalgo Otamendi, Antonio 1 ; Fernández, Jose Francisco 2 1 Fundación CIDAUT; Parque tecnológico de Boecillo, P.209, Boecillo. Valladolid. pain; migmor@cidaut.es; beaces@cidaut.es; robcor@cidaut.es; anthid@cidaut.es 2 Instituto de Cerámica y Vidrio, C/Kelsen, 5. Madrid pain; jfernandez@icv.csic.es ABTRACT In this paper a design methodology of components with piezoelectric elements, as active or movement generators, is presented. In recent years the development of smart structures that use piezoelectric sensors and actuators has increased to obtain the desired mechanical behaviour. imulation is a widely used tool in the designing of products that allows a better understanding of the model behaviour, and a reduction of the number of prototypes. This leads to a reduction in costs and development time, and increases competitiveness. The Acoustics and Vibrations Department of the CIDAUT Foundation along with the lectroceramics Department of ICV have developed a methodology for the simulation of piezoelectric elements that takes into account their electrical and mechanical properties using a general finite element software. Numerical results have been correlated with experimental results. INTRODUCTION imulation tools allow us to predict structural behaviour from a design (shape, dimensions, material properties ). A design can be discretized into elements where different behaviour laws can be applied according to the physical phenomena to study. Besides the prediction of structural behaviour, simulation tools are used to optimize a design, with the objective of minimizing/maximizing a response such as velocity vibration in a point, reducing mass imulation codes used in mechanical behaviour are more and more versatile, and allow taking into account electrical, acoustical, thermal behaviour and different couplings. Traditional codes used to solve coupled electric problems work in 2D domains, whereas in mechanical simulation powerful tool are used to calculate complex 3D structures with millions of d.o.f.s. Next how to study piezoelectric behaviour coupled to structural vibrations will be shown from the thermal behaviour model using the commercial software MC.Nastran. A cymbal structure made of a piezoelectric disc with a preformed brass sheet on each side will be modelled using this method. This approximation allows studying complex structures which require a high computational power. THORY FOR PIZO LCTRIC CRAMIC IMULATION When using the finite element method to solve a mechanical problem, first it is necessary to mesh the structure in elements where basic behaviour laws can be applied and certain hypothesis are assumed. The following equations describe piezoelectric behaviour:

2 ε = s σ d D = dσ + p t q. 1 Where σ are mechanical stresses, ε are strains, represents the electric field, D is the electric displacement, s is the mechanical compliance matrix for a constant field, p is the permittivity matrix for a fixed state of deformation, and d is the piezoelectric matrix. The electric field is related to the electrical potential φ by: = gradφ q. 2 And the mechanical strain ε to the mechanical displacement by: ε = Bu q. 3 Developing these equations and applying Maxwell and classical mechanic laws R.Lerch [1] obtains: Mu&& + D K t uφ uu u + K u& + K φφ uu φ = Q u + K + Q uφ P φ = F B + F + F p q. 4 Where u is the displacement vector, φ is the electric potential, M is the mass matrix, D is the damping matrix, K uu is the mechanical stiffness matrix, K uφ is the piezoelectric coupling matrix, K ΦΦ is the dielectric matrix. F B are mechanical volume forces, F are mechanical surface forces, F P are mechanical point forces, Q are surface electrical charges and Q P point charges. If we consider the electric potential φ known, the system of equations is reduced to the first equation (in q. 4) where the variable to solve is displacement u. MC/NATRAN, one of the most widely used finite element solver, offers no specific element to model piezoelectric structures directly. Rather, the analogy between piezoelectric strain and thermally induced strain is used. Piezoelectric coefficients characterizing the actuator are input as thermal expansion coefficients, and the voltage as a temperature increment. This formulation may lead to the resolution of the problem using the matrix formula implemented in MC.Nastran for MAT9 anisotropic material: σ 11 σ 22 σ 33 = τ 12 τ 23 τ ε 11 A1 ε 22 A2 ε 33 A3 γ 12 A4 γ 23 A5 γ 13 A6 ( φ φ ) ref q. 5 Next the methodology used to implement piezoelectric material properties in this formulation to solve the coupled mechanical electric problem is discussed. 19 th INTRNATIONAL CONR ON ACOUTIC ICA2007MADRID 2

3 MATRIAL CHARACTRIZATION The following table shows the piezoelectric material properties obtained from tests and bibliography: Table 1. Piezoelectric material properties PZT 5A 11 (m 2 /N) (m 2 /N) (m 2 /N) lastic constants 33 (m 2 /N) (m 2 /N) (m 2 /N) Density (kg/m 3 ) Dielectric constants Piezoelectric constants d 15 (m/v) ε 11 /ε 0 d 31 (m/v) ε 33 /ε 0 d 33 (m/v) d 24 (m/v) APPLICATION XAMPL As an application example a finite element model of a cymbal is shown. The objective is to optimize the brass structure to obtain maximum response amplification. The cymbal consists of a piezoelectric disc 1mm thick and 12mm in diameter. On each side of the disc a 0.3mm thick stamped brass sheet is bonded with a 0.02mm thick epoxy lamina. The following figures show the test and finite element models. Figure 1. Cymbal test model 19 th INTRNATIONAL CONR ON ACOUTIC ICA2007MADRID 3

4 8.7m m 4.7m m Figure 2. Finite element model The finite element model consists of 16,368 solid elements and 20,611 nodes. The following hypotheses have been posed: Linear behaviour mall displacements and strains Piezoelectric material with anisotropic behaviour A frequency response analysis is made for a constant electric potential 10V in a frequency range up to 2000Hz. Figure 3 shows the response of a point in the upper sheet of the cymbal structure: Figure 3. imulated frequency response The calculation of a model like this takes only a few minutes. In order to validate this simulation methodology, the vibration of the metallic structure has been measured when an electric potential is applied to the piezoelectric disk. Figure 4 compares experimental and simulated results: 19 th INTRNATIONAL CONR ON ACOUTIC ICA2007MADRID 4

5 Figure 4. xperimental (red) and simulated (blue) responses The resonance frequency for the simulated model is slightly lower than the measured one. The damping in the simulated model needs to be updated. CONCLUION This paper shows how to simulate coupled electrical-mechanical behaviour with commercial structural software. Although lacking final tuning of the model, a methodology has been presented to obtain validated and robust results. This analysis tool is quite powerful, and allows the simulation of complex structures with low calculation times. References [1] Reinhard Lerch. imulation of Piezoelectric devices by two and three dimensional finite elements. I Transactions on Ultrasonic, ferroelectric and frequency controls Vol 37 N2 May 1990 [2] P. Ochoa, M. Villegas, J.L Pons, P. Ledinger, J.F, Fernández. Tunability of cymbals as Piezocoposite transducers. Journal of lectroceramics. February 26, th INTRNATIONAL CONR ON ACOUTIC ICA2007MADRID 5